Caramelize the onions. Melt the butter with the olive oil in a large oven-safe skillet or Dutch oven over medium heat. Add the sliced onions, sugar, and salt.
Cook for 20–25 minutes, stirring occasionally, until the onions become soft and deeply golden. Lower the heat if they begin browning too quickly. Add the minced garlic during the final 1–2 minutes.
Toast the orzo. Stir the dry orzo into the caramelized onions and cook for about 2 minutes, stirring frequently.
Add the chicken and seasonings. Stir in the shredded chicken, dried thyme, black pepper, and optional balsamic glaze until evenly combined.
Add the liquids. Pour in the chicken broth and heavy cream. Stir well and bring everything to a gentle simmer.
Reduce the heat to low, cover, and cook for approximately 8–10 minutes, stirring occasionally so the orzo doesn't stick to the bottom. Cook until the pasta is nearly tender and the mixture is creamy.
Add the cheese. Stir 1 cup of mozzarella and all of the Parmesan into the French Onion Chicken Orzo Casserole until melted and creamy.
Bake. Preheat the oven to 375°F (190°C). Sprinkle the remaining 1/2 cup mozzarella evenly over the casserole.
Bake uncovered for 10–15 minutes, or until the sauce is bubbling and the cheese is melted and lightly golden.
Rest and serve. Let the casserole rest for 5–10 minutes before serving. Garnish with fresh thyme or parsley if desired.